Where c1 and c2 are constants. This is a model of a physical process and it was shown that z dimension is quantified by a chunks of a constant size h and x is monotonously increasing, so $\frac{\partial x}{\partial z}=\frac{\Delta x}{\Delta z}=\frac{x_i - x_{i-1}}{h}$

First of all, you can already solve for R(t) by hand. Second, the remaining equation probably should have C replaced by C_i. Is that what you mean? You will then also need the initial conditions for all C_i, I would guess. With that, you would have a coupled system of first-order equations in time for the C_i.
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JensMay 19 '12 at 4:50

I've updated the question. Wb is a function dependent on C, I'm not sure if it possible to solve it by hand.
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AndrewMay 19 '12 at 14:20

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So, initially $x=x(y,z)$ but you would like to replace this 2d function by a finite number of 1d functions, ie, you'd like to discretise the $z$ direction by hand. Right?
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aclMay 19 '12 at 17:07

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@Andrew then you need to solve a set of differential equations for $x_i(y,t)$ (ie, a set of variables, not a single var) and $y(x_1,\ldots,x_N,t)$, whereas you're trying to set the problem up for a single $x$.
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aclMay 19 '12 at 17:30

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Look up Delayed Differential Equations in the documentation (howto/SolveDelayDifferentialEquations). Your ?? will be x[t-h]. Also figure out the correct initial condition because you will need to specify it for a range of values.
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Daniel LichtblauMay 20 '12 at 16:10

2 Answers
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It seems (from the comments) that what you want to do is this: initially x=x(y,z) but you would like to replace this function of 2 vars by a finite number of functions of 1 var, ie, you'd like to discretize the $z$ direction by hand. This means that you need to solve a set of differential equations for the $x_i(y,t)$ (a set of functions, not a single fun) and $y(x_1,\ldots,x_N,t)$.

You're trying to set the problem up for a single $x$, and that is the problem.

OLD ANSWER

(I think this is neat so I'll leave it here for now)

I very likely misunderstood you. If, however, I understood correctly the question, it is something like: Suppose I have $y'(t)=f(y)$ and ask mathematica to solve it numerically. How do I inspect the values of $t$ used?

The answer is to rig the ODE so that you can peek at the values mathematica evaluates, as follows:

Your syntax has me a bit confused. SetDelayed (:=) looks off to me. Typically one uses this to define functions. I think you want Equal (==) where you have the system of equations and Set (=) where you assign the value of the expression to "solution".

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